How do I align the view to the local axis of an object?
Press Shift in combination with any of those number pad numbers to align to the local axes of the selected object.

Note that this will only work if you have rotated your object in object mode and haven't applied the rotation. If you have rotated in edit mode or applied the rotation then you can select a face (or vertices or edges) of the object and use Shift+1 (3,7,9 etc.) to align the view relative to the axis of the selected element.

If you use Ctrl+Shift in combination with the numpad numbers you will align to the opposite view of the object's axes, e.g. bottom, back instead of top, front.